Extension of Gyárfás-Sumner Conjecture to Digraphs

Abstract

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. 
 Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

Heroes in orientations of chordal graphs

“…Given a digraph H, denote by F orb ind (H) the class of digraphs with no induced copy of H. A result of [11] implies that if H is not an orientation of a forest, then no digraph is a hero in F orb ind (H) except for the isolated vertex and the arc. A systematic study of heroes in classes of digraphs of the form F orb ind (H) where H is an oriented forest has been initiated in [4]. An oriented star is an orientation of a star, that is a tree with only one non-leaf vertex.…”

“…An oriented star is an orientation of a star, that is a tree with only one non-leaf vertex. It is proved [4] that if H is not the disjoint union of oriented stars, then no hero in F orb ind (H) contains a directed triangle. A result in [7] implies that every transitive tournament is a hero in F orb ind (H) when H is a disjoint union of oriented stars.…”

“…Concerning superclasses of chordal graphs, consider the following construction (already mentioned in [4]). Let G 1 be the graph on 1 vertex, and having defined G k−1 inductively, define G k as follows: start with three disjoint copies G…”

Heroes in oriented complete multipartite graphs

“…Observe that if a class of digraphs C contains all tournaments, then a hero in C must be a hero in tournaments. In [3], it is conjectured that heroes in oriented complete multipartite graphs are the same as heroes in tournaments (actually a wider conjecture is proposed, see Section 5). We disprove this conjecture by showing the following:…”

“…of a star). In [3], the authors initiated a systematic study of heroes in F orb ind (F ) for a fixed digraph F . We now summarize the known results in this direction and explain how our results fit in the big picture.…”